How can I use Generative Adversarial Networks to solve the imbalanced class problem?
In my experience, GANs work really well for the scenario of semi-supervised learning, where you don't necessarily have labels for all your class $B$ data, but you do have a balanced dataset. In my (limited) experience, you do have to have a balanced (in numbers) set of $A$ and $B$ objects, even though you are not sure of the labels.
And yes, GANs can overfit to outliers as well, especially in the absence of a lot of examples, so be cautious.
Currently, the version that works best for me (in terms of GANs) is WGAN-GP or WGAN-LP in combination with Optimistic Mirror Descent Adam (here, Ncritic/Nactor = 1). Take a look at the paper by Adiwardana et al., especially Fig. 7 (astonishing!), for semi-supervised learning with a limited number of class labels.